List of practice Questions

For g ∈ \(\Z\), let \(\bar{g}\) ∈ \(\Z_8\) denote the residue class of g modulo 8. Consider the group \(\Z^×_8\) = {\(\bar{x}\) ∈ \(\Z_8\) : 1 ≤ x ≤ 7, gcd(x, 8) = 1} with respect to multiplication modulo 8. The number of group isomorphisms from \(\Z^×_8\) onto itself is equal to ________

Let f(x) = \(\sqrt[3]{x}\) for x ∈ (0, ∞), and θ(h) be a function such that
f(3 + h) − f(3) = hf′ (3 + θ(h)h)
for all h ∈ (−1, 1). Then \(\lim\limits_{h→0} θ(h)\) is equal to _________. (rounded off to two decimal places)

Let V be the volume of the region S ⊆ \(\R^3\) defined by
S = {(x, y, z) ∈ \(\R^3\) : xy ≤ z ≤ 4, 0 ≤ x² + y² ≤ 1}.
Then \(\frac{V}{π}\) is equal to ________ . (rounded off to two decimal places)

The sum of the series \(\sum\limits_{n=1}^{\infin}\frac{2n+1}{(n^2+1)(n^2+2n+2)}\) is equal to _________.(rounded off to two decimal places)

The value of \(\lim\limits_{n \rightarrow \infin}\left(1+\frac{1}{2^n}+\frac{1}{3^n}+...+\frac{1}{(2023)^n}\right)^\frac{1}{n}\) is equal to _____________ . (rounded off to two decimal places)

Let y : (1, ∞) → \(\R\) be the solution of the differential equation
\(y"-\frac{2y}{(1-x)^2}=0\)
satisfying y(2) = 1 and\(\lim\limits_{x→∞}y(x) = 0\). Then y(3) is equal to __________. (rounded off to two decimal places)

Let S be the triangular region whose vertices are (0, 0), \((0,\frac{\pi}{2})\) and \((\frac{\pi}{2},0)\). The value of \(\iint\limits_S\sin(x)\cos(y)dx\ dy\) is equal to ________. (rounded off to two decimal places)

Let W be the subspace of \(M_3(\R)\) consisting of all matrices with the property that the sum of the entries in each row is zero and the sum of the entries in each column is zero. Then the dimension of W is equal to ___________.

The maximum number of linearly independent eigenvectors of the matrix
\(\begin{pmatrix} 1 & 1 & 0 & 0 \\ 2 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 1 & 3 \\ \end{pmatrix}\)
is equal to _________.

Let S be the set of all real numbers α such that the solution y of the initial value problem
\(\frac{dy}{dx}=y(2-y),\\y(0)=\alpha,\)
exists on [0, ∞). Then the minimum of the set S is equal to __________. (rounded off to two decimal places)

A competitive firm can sell any output at price 𝑃 = 1. Production depends on capital alone, and the production function 𝑦 = 𝑓(𝐾) is twice continuously differentiable, with
𝑓(0) = 0, 𝑓 ′ > 0, 𝑓 ′′ < 0, \(lim\\_{ 𝐾→0 }\) 𝑓 ′ (𝐾) = ∞ , \(lim\\_{ 𝐾→∞}\) 𝑓 ′ (𝐾) = 0.
The firm has positive capital stock 𝐾̅  to start with, and can buy and sell capital at price 𝑟 per unit of capital. If the firm is maximizing profit then which of the following statements is NOT CORRECT?

Let 𝑓, 𝑔∶ℝ → ℝ be defined by

Then

Let 𝑆 be a feasible set of a linear programming problem (𝑃). If the dual problem of (𝑃) is unbounded then

Which of the following is NOT CORRECT?

Among the following statements which one is CORRECT?
S1: 𝑥²+𝑦²=6 is a level curve of 𝑓(𝑥, 𝑦)=\(\sqrt{𝑥^2+𝑦^2}\)−𝑥²−𝑦²+2 
S2: 𝑥²−𝑦2=−3 is a level curve of 𝑔(𝑥, 𝑦) =\(𝑒−{^𝑥}^{2} 𝑒{^y}^{2}\)+𝑥⁴−2−2𝑥²𝑦²+y⁴

Which of the following is NOT a component of Gross Domestic Product?

Which of the following are the direct instruments exercised by the Reserve Bank of India to control the money supply?
(i) Cash Reserve Ratio 
(ii) Open Market Operations 
(iii) Foreign Exchange Rate 
(iv) Statutory Liquidity Ratio

Which of the following committees for the first time recommended for India
(i) use of implicit prices derived from quantity and value data collected in household consumer expenditure surveys for computing and updating the poverty lines 
(ii) Mixed Reference Period (MRP) in estimating poverty lines

Which of the following Five Year Plans focused on rapid industrialization-heavy and basic industries, and advocated for a socialistic pattern of society as the goal of economic policy?

Let 𝑀 and 𝑁 be events defined on the sample space 𝑆. If 𝑃(𝑀) = \(\frac{1}{3}\) and 𝑃(𝑁^𝑐 ) = \(\frac{1}{4}\) then which one of the following is necessarily CORRECT?

Consider a 2-agent, 2-good exchange economy where agent 𝑖 has utility function 𝑢𝑖 (𝑥𝑖 , 𝑦𝑖 ) = max{𝑥𝑖 , 𝑦𝑖 }, 𝑖 = 1,2. The initial endowments of goods 𝑋 and 𝑌 that the agents have are (𝑥̅̅1̅, 𝑦̅̅1̅, 𝑥̅̅2̅, 𝑦̅̅2̅) = (25, 5, 5, 5). Then select the CORRECT choice below where the price vector (𝑝𝑥, 𝑝𝑦) specified is part of a competitive equilibrium.

For a firm operating in a perfectly competitive market which of the following statements is CORRECT?

A firm is operating in a perfectly competitive environment. A change in the market condition leads to an increase in the firm’s profit by an amount K. Which of the following describes the change in the Producer’s Surplus due to the above change in the market condition?

Two firms, X and Y, are operating in a perfectly competitive market. The price elasticity of supply of X and Y are respectively 0.5 and 1.5. Then

Let 𝑦 = 𝑦(𝑥) be a solution curve of the differential equation
\(𝑥 \frac{𝑑𝑦}{d𝑥} = 𝑦 \) ln ( \(\frac{y}{x}\) ) , 𝑦 > 𝑥 > 0. 
If 𝑦(1) = 𝑒 ² and 𝑦(2) = 𝛼, then the value of \(\frac{𝑑𝑦}{𝑑𝑥}\) at (2, 𝛼) is equal to